models, metrics, and an index to assess humanitarian response

Models, Metrics, and an Index to AssessHumanitarian Response Capacity

Jason AcimovicSmeal College of Business, The Pennsylvania State University, [email protected]

Jarrod GoentzelCenter for Transportation and Logistics, The Massachusetts Institute of Technology, [email protected]

February 20, 2015

The race to meet vital needs following sudden onset disasters leads response organizations to establish stock-piles of inventory that can be deployed immediately, without needing to rely on suppliers. These governmentor non-government organizations dynamically make stockpile decisions in isolation, and they do so withoutempirical metrics on how the stock will improve the response effort. To address these issues we develop aquantitative model to analyze stockpile capacity, propose metrics based on model results to assess decisionsover time, and normalize the metrics as the basis for a sector-wide index of combined capacity. To guide thedevelopment of our analytical assessment approach, we utilize comprehensive information on global stockincluding publicly available data on inventory stored by various organizations in United Nations facilitiesand data that organizations provide to the UN on their own stockpiles. In addition, our empirical studyoffers practical insights regarding the current humanitarian response capabilities and strategies. We showthat significant improvement in terms of time and cost can be achieved by reallocating the inventory ofseveral critical items currently in the system and coordinating among the organizations.

Key words : Humanitarian logistics, Pre-positioning, Stockpiling, Logistics index, Disaster Response

1. Introduction

The capability to rapidly deploy life-saving commodities in response to natural disasters is vital.

Government and non-government organizations have sought to improve response capacity by

procuring larger stockpiles of critical commodities and pre-positioning them in various locations

prior to disaster events. In addition, the private sector is increasingly making commitments to

donate commodities following a disaster, which effectively adds to the stockpile. While the efforts

of numerous organizations to increase stockpiles certainly improve capacity, the overall impact of

larger and more dispersed stock deployment on humanitarian response is difficult to assess.

Our research seeks to address two issues in response capacity assessment: (1) the fragmented

approach to establishing stockpiles makes it difficult to consider the combined capacity, and (2) the

humanitarian sector lacks models and metrics to assess the quality of the stock positions over time.

To address these issues we develop a quantitative model to analyze stockpile capacity, propose

metrics based on model results to assess decisions over time, and normalize the metrics as the basis

for a sector-wide index of combined capacity.

Previous models for humanitarian prepositioning and stockpiling have focused on determining

optimal strategies for facility location and stock deployment. Our modeling objectives differ from

1

Acimovic and Goentzel: Models, Metrics, and an Index to Assess Humanitarian Response Capacity2 Article submitted to Journal for Peer Review

these efforts in two aspects. First, we do not emphasize the stockpile location decision. Often

humanitarian warehouse facilities are already established or determined by the donor that subsi-

dizes the effort. Thus, we focus on allocation of stock across a set of given locations. Second, we do

not seek to prescribe optimal solutions, but rather to measure the quality of current and alternate

strategies compared with optimality. Like the previous approaches, we use a two-stage stochastic

linear program (LP) to determine optimality based on detailed forecasts of humanitarian needs.

Next, we develop response capacity metrics that assess stockpile decisions based on robust data

provided by the stochastic model. The metrics consider the overall capacity of stocks to meet

the needs of the affected population and the quality of stock allocation considering time and cost

for deployment to disaster locations. These metrics can be utilized by organizations to evaluate

strategies such as global stock level, by considering the marginal value of investment, and stock

allocation, by considering the marginal value of repositioning. The model enables robust measure-

ment considering aspects such as cutoff times for delivering aid and seasonality of natural disasters

throughout the year. In addition to individual metrics, we can develop the efficient frontier for stock

deployment considering the primary objectives of minimizing response time and transportation

cost.

Finally, we normalize these metrics and propose an initial sector-wide Response Capacity Index

(RCI) that considers stock deployment decisions across organizations. The index should be updated

regularly and its results be made available publicly. The RCI can be used as a dynamic managerial

tool by organizations seeking to deploy resources that address gaps in the combined capacity.

Donors can use it to assess investment strategies. It also offers transparency and accountability

to the public regarding resources established for their benefit. Much like industry benchmarking

efforts, we expect that the value of dynamic information on combined capacity is sufficient incentive

for organizations to share data on a regular basis.

We test our model, metrics, and index with extensive data on commodity stock levels provided

by the United Nations (UN). The data include stock in global facilities managed by the UN and by

organizations that shared their data with the UN. These data provide the most accurate picture

of commodities established to respond to disasters anywhere in the world. Thus, in addition to

introducing a new approach to better assess capacity, our empirically grounded modeling work

offers practical insights regarding the current humanitarian response capabilities and strategies.

Our results show that significant improvement can be achieved simply through reallocating the

current inventories across depots: the current allocation takes 14% longer and costs 29% more on

average than the respective optimal allocation. We also learn which locations are more important

with different levels of stock and at different times of the year. Dubai is one of the most important

Acimovic and Goentzel: Models, Metrics, and an Index to Assess Humanitarian Response CapacityArticle submitted to Journal for Peer Review 3

depots when inventory levels are low; Subang, Malaysia, becomes increasingly important as inven-

tory levels increase due in part to the large disasters that strike the Pacific region. Results help

build new intuition, such as Stockholm serving as a useful depot for blankets in the summer due

to its relatively short flight paths to Asia. Dual variables quantify the value for decision makers,

in time or cost saved, of shifting stock to better locations.

The rest of the paper is organized as follows. We describe the humanitarian context that moti-

vates this research in section 2. In section 3 we discuss the relevant literature, specifically related

to stochastic optimization in pre-positioning supplies and indices in the realms of logistics and

disasters. We introduce our analytical approach and stochastic LPs in section 4, followed by the

metrics we propose based on model outputs. The empirical data for our study are presented in

section 5, including the demand, supply, delivery times and costs, and commodity specifics such

as people-served-per-item and weight. Section 6 presents the results of utilizing the data in section

5 as inputs into the models in section 4. We outline how the resulting metrics might provide the

foundation for an index in section 7. Finally, section 8 concludes the paper.

2. Context and Motivation

Immediately following a disaster that outpaces community coping mechanisms, various outside

organizations rush to provide life-saving commodities to meet health, water, food, shelter, or other

needs for the affected population. The race to meet urgent needs relies on commodity stocks that

have already been prepositioned by these organizations, which could include government (local,

regional, national, or foreign), non-government (NGO), military, or private sectors. The stock for

this initial deployment could be centralized or deployed across several locations. For large-scale

and/or urgent crises, organizations may choose to utilize several stock locations and incur the

additional cost of shipping farther to meet needs. In most cases, this initial stock is intended to meet

human needs within the first few days, followed by replenishment from strategic suppliers based

on assessments of need in the affected community. Hence, the initial push is typically transported

by air unless ground transportation offers a 1-2 day transit time from a nearby stocking point; sea

shipping is only used for replenishment.

Poor humanitarian response performance to a widely publicized event pushes organizations to

take tangible actions; often this results in increasing the size and/or number of locations for critical

commodity stockpiles (typically these are skewed toward the nature of the recent event and do not

consider broader risks). On the other hand, constrained fundraising and/or expiration of stockpiled

items pushes organizations to reduce stock. These continual adjustments occur across a very frag-

mented humanitarian sector with numerous NGOs and government agencies independently taking

actions for the same population. The fragmented nature of the sector and limited transparency


about immediate response capacity makes it difficult to assess the level of preparedness for a region.

Both feast and famine have negative outcomes: overcapacity leads to material convergence that

stresses bottleneck resources following a disaster or to product expiry if there are few disasters

requiring the stock; undercapacity leads to unmet needs in a large crisis.

Our research is motivated by these shortcomings. First, rather than address the decision mak-

ing for a particular organization, we aim to facilitate sector wide response capacity planning by

considering combined resources. Six United Nations Humanitarian Response Depots (UNHRDs)

around the world Accra, Brindisi, Dubai, Las Palmas, Panama, and Subang offer space for gov-

ernment and non-governmental organizations to stockpile disaster relief supplies. These services

are offered at no-cost or on a cost-recovery basis. In addition, the UN Office for the Coordination

of Humanitarian Affairs (OCHA) manages a “Global Mapping of Emergency Stockpiles” to track

items stocked by various organizations in their own facilities. In OCHA’s own words (UN Office

for the Coordination of Humanitarian Affairs 2014a): “It provides a single-entry interface to map

the capacities and resources of humanitarian actors to respond to the needs of affected populations

in case of emergency. It is a central platform that places increased emphasis on ‘who has what

were’ by region, sector, organization and/or organization type.” Participation is voluntary and each

organization maintains its own data. As such, it does not reflect all the items available to deploy

by the global disaster relief community and it may not be current. However, as far as we know, it

is the most comprehensive and accurate database describing global stockpiles. These UN sources

provide the best data to represent combined capacity.

Second, instead of basing decisions on recent events and short memories, we aim to assess capacity

on a broader base of information. We assess stockpile capacity considering sudden onset disasters

and epidemics going back more than two decades. The Centre for Research on the Epidemiology of

Disasters (CRED) manages the Emergency Events Database (EM-DAT) that records data on the

occurrence and effects of over 18,000 mass disasters in the world from 1900 to present (Centre for

Research on the Epidemiology of Disasters 2014). Despite the limitations of any database with the

objective of recording details related to every disaster that occurs (Guha-Sapir and Below 2006),

this database is recognized as the best of its kind and has been utilized by many other researchers

working in similar domains (Peduzzi et al. 2009, Duran et al. 2011, 2013). As our research is

not focused on developing forecasts, our study assumes a future that is similar to the past; more

specifically, we assume that each disaster recorded between 1990 until the summer of 2013 has an

equal chance of occurring again. Beyond this study, our modeling framework generalizes to consider

any proposed disaster forecast.

Our analytical assessment approach – including the model, metrics, and index – is developed

in the context of these data. Our empirical study also demonstrates the usefulness of considering

combined resources and a broad base of disaster events in planning humanitarian response capacity.


3. Literature Review

Our work is mainly related to two streams of literature. The first stream utilizes stochastic opti-

mization to pre-position supplies for disasters. The second stream defines indices in the realms of

logistics as well as disaster vulnerability and preparedness.

The set of metrics and tools we develop is based on a two-stage stochastic linear program. Several

authors have used this type of model to optimize placement of inventory in order to respond to

disasters. Many of these models determine where to place inventory in the first stage in order to

optimize the response in the second stage. The second stage of these stochastic linear programs

is often represented by a set of disaster scenarios. The paper most related to our work is that of

Duran et al. (2011). The authors work with CARE International to decide which depots to open

and how much to store in each, with the objective of minimizing average time to respond to a

disaster with a C-130 aircraft. Similar to this work, they utilize Em-Dat data (Centre for Research

on the Epidemiology of Disasters (2014)) in their scenario generation, measuring the total affected

population for each disaster. They allow multiple disasters to occur within the replenishment lead

time (so that a scenario may include more than one disaster). While they consider factors that we

omit (multiple disasters, need for items based on disaster type, etc.), we consider factors they omit,

such as transportation by truck and transportation costs in general (in addition to time); we also

use the results of the stochastic linear program to develop a normalized set of metrics, as opposed

to using it solely to determine where to place items. Other work in this area includes the following

papers. de Brito Junior et al. (2013) develop a model that minimizes unmet demand as well as

costs, and that takes into account donations based on media response. They apply their model to

scenarios in Brazil. In Mete and Zabinsky (2010), the authors analyze where to place supplies in

Seattle in order to respond to possible earthquakes. They take into account possible road closures

as well as the traffic situation (rush-hour, weekend, weekday). They minimize multiple objectives

of cost/time and unmet demand, whose relative weights are determined by a tuning parameter.

Klibi et al. (2013) develop richer sets of scenarios that account for where the disaster is, how it

unfolds over time to neighboring regions, what the demand is, how vendor capacity may be limited,

and how depots may be inaccessible. Because the complexity associated with the richer set of

demand scenarios, the authors use Monte Carlo simulation to generate sample paths. Especially,

de Brito Junior et al. (2013) and Klibi et al. (2013) have excellent literature reviews. Salmern and

Apte (2010) propose a multi-objective stochastic optimization problem that dictates how best to

allocate a budget in order to minimize casualties. In Hong et al. (2015), the authors formulate a

stochastic program that takes into account reliability; probabilistic constraints are included that

ensure commodities can be delivered with high probability. Beyond the above literature, there has


been much work in stochastic programming for general operations problems outside of disaster

relief. See, for example, Shapiro et al. (2014) for an introduction to stochastic programming.

The second stream of literature related to our work consists of papers that define indices. Within

this stream, we highlight indices developed for the logistics community and indices developed for

the disaster relief community. Several indices developed for the logistics community are created

by international organizations such as the World Bank. Arvis and Shepherd (2011) develop an air

connectivity index based on gravity model theory from economics. Hoffmann (2012) develop an

index to measure port performance: the Liner Shipping Connectivity Index (LSCI). This index

is based on the normalized average of five components. Lastly, the Logistics Performance Index

(Arvis et al. (2014)) is based on survey results of freight forwarders. From the questions on the

survey, Principle Component Analysis (PCA) is used to weigh the survey question responses into a

single index number. Our work adds to the logistics index literature by using linear programming

as an analytical tool for metrics that form an index. Acimovic and Graves (2015) use a normalized

(non-stochastic) linear program to develop an inventory balance metric for an online retailer. Some

of our work is based on that balance metric. However, we account for demand stochasticity, which

is not included in Acimovic and Graves (2015).

Many authors have written papers describing indices related to disaster preparedness and vul-

nerability. Simpson and Katirai (2006) provide an excellent overview of these indices, describing

many of them in detail in their appendix. Many of these indices measure risk to different communi-

ties, where risk might be some combination of vulnerability and response. For instance, Davidson

and Lambert (2001) develop a multiplicative index that has two variants: measuring economic risk

and measuring life risk. The index includes: hazard, exposure, vulnerability, and response. The

response factor includes: percent of the country detached from the mainland, number of shelters

available, evacuation clearance time, percent expected to evacuate, population density, city lay-

out (grid or not), number of hospital beds per 100,000 people, number of physicians per 100,000

people, and per capita state gross product. The weights of these components in determining the

response factor are found through the analytical hierarhcical process (AHP). The National Health

Security Preparedness Index (NHSPI) (Association for State and Territorial Health Officials and

Centers for Disease Control and Prevention (2014)) is a coordinated effort between the Association

of State and Territorial Health Officials (ASTHO) and the Centers for Disease Control (CDC).

The index comprises 5 domains and 14 subdomains. The 14 subdomains consist of 128 normal-

ized measures. Specifically related to our work is the subdomain “Medical Material Management,

Distribution, and Dispensing.” It is the simple average of ten measures relating a state’s ability

to “acquire, maintain, ..., transport, distribute, and track medical materiel ... before and during

an incident and recover an account for unused medical materiel after an incident.” This index is


updated regularly, and its results (of all domains, subdomains, and measures) are available publicly

(http://www.nhspi.org/). Our end goal is to make our index and sets of metrics available in real

time to the public as well. A UN disaster Risk Index (described in Peduzzi et al. (2009)) assesses

which countries are most at risk to hazards such as droughts, floods, cyclones, and earthquakes. It

uses Em-Dat data (Centre for Research on the Epidemiology of Disasters (2014)) for a regression

model that predicts total persons killed based on disaster type, gross domestic product (GDP),

number of people living in watersheds (for floods), percent of country’s land that is arable (for

droughts), and others. This is combined with population levels and disaster frequency to define

overall risk per country.

Many of the indices in this stream are based on economics models. They utilize PCA, regression,

or simple or weighted averages of objective and subjective measures. In contrast to this, we develop

a set of objective metrics based partially on the output of linear programming models.

In the disaster response literature, stochastic linear programming has been used to decide where

to position items, and logistics and disaster indices have been based on economics or averaging

models. We contribute to the literature by applying the techniques of stochastic linear programming

in the development of normalized, useful, objective metrics. It is our hope to build upon these

metrics and further develop our preliminary index with the ongoing help and feedback from the

disaster response community.

4. Analytical approach

Our analytical approach is based on a stochastic linear program and various metrics based on

model results. They are described in detail below.

4.1. Formulation of the stochastic linear program

One of the building blocks for the set of metrics and the index is a scenario-based stochastic

linear program (LP). The stochastic LP has two stages. Depending on whether we are utilizing

actual or optimal inventory allocations, the first stage merely records the allocation of inventory or

determines where to place inventory. The second stage is a transportation problem that allocates

the resulting supply to demand, where the demand occurs at a single disaster node in each scenario.

The stochastic LP minimizes either the total expected time or cost to deliver the items from the

supply nodes to the disaster locations across the scenarios.

Some of the output elements we utilize from this stochastic LP are the objective value, normalized

objective value, and dual variables.

We first define parameters and variables that the stochastic LP uses. A dummy supply node is

employed in order to satisfy demand for disasters when the need exceeds what is on-hand in the


depots. We also include a delivery deadline parameter θ in order to examine only those solutions

able to serve beneficiaries within a given time window.

I 3 i−Set of all depots and warehouses except the dummy node

iW −The dummy supply node

IW ≡ I ∪ iW

−Set of all depots including dummy node

K 3 k−Set of possible disaster scenarios

J 3 j−Set of possible disaster locations

M 3m−Set of disaster types

N 3 n−Set of line items

R 3 r−Set of transportation modes

cijnr−Cost in dollars to transport a single item n from i to j via mode r

τijr−Time to ship a single item from i to j via mode r (item independent)

cW (τW )−The cost (time) from the dummy supply node to a disaster

pk−Probability of scenario k occuring(∑

pk = 1)

Sjmnt−The domestic/local capacity to respond to a disaster in location j of type m

at period t for item n

X−The inventory vector dictating how many items are stored at each depot i. Xi’s

are the elements of this vector.

χ∈N−Starting inventory in the system as a whole, not including the dummy node

TAP k−Total Affected Population in scenario k

βjmnt−Factor converting number of people affected into the demand for item n at

location j at time t for disaster type m

jk−Location of disaster k

dkn ≡max(TAP kβjk,m,n,t−Sjk,m,n,t,0

)−Actual demand for item n for disaster k

ykinr−Decision variable for how much of n to send from i to the disaster in scenario

k via mode r. (Note we have n in the subscript, but for now we decompose

the problem by item, so it is redundant)

X−The |I| dimensional vector of starting inventory in each supply node. Its elements

are Xi. (Depending on the formulation, this may be and input by the user

or a decision variable

θ−Delivery deadline: if θ <∞, arcs whose τijr > θ are removed, for both cost

and time objectives.

The stochastic LP formulation for minimizing time is (note: the LP that minimizes cost is

analagous):

V W (X, n)≡ miny

∑k

pk∑

i∈IW ,r

τi,jk,rykinr (1a)


s.t.∑

i∈IW ,r

ykinr = dkn ∀k (1b)

∑r

ykinr ≤Xi ∀i∈ I, k (1c)

ykinr ≥ 0 ∀i, k, r (1d)

We recognize that one could solve the above problem without a stochastic LP, namely, because

the only decision variables are how much to ship from each depot to each disaster and because we

assume there is only one location affected per scenario. However, keeping it in the form of a linear

program maintains flexibility if more than one disaster were to be considered per scenario, allows

us to calculate dual variables easily, and lends itself the the following formulation in which we

calculate the optimal allocation of inventory. The formulation that optimizes inventory allocation

is very similar, with X as a decision variable, and with an additional constraint to ensure the sum

of the supply is equal to χ.

V OPT,W (χ,n)≡ minX,y

∑k

pk∑

i∈IW ,r

τi,jk,rykinr (2a)

s.t.∑

i∈IW ,r

ykinr = dkn ∀k (2b)

∑r

ykinr ≤Xi ∀i∈ I, k (2c)∑i∈I

Xi = χ (2d)

ykinr ≥ 0 ∀i∈ IW , k, r (2e)

Xi ≥ 0 ∀i∈ I (2f)

We define the objective values with the dummy costs subtracted as:

V (X, n)≡ V W (X, n)−∑k

pk∑r

τiW ,jk,rykiW ,n,r

V OPT (χ,n)≡ V OPT,W (χ,n)−∑k

pk∑r

τiW ,jk,rykiW ,n,r

where the ykiW ,n,r

’s are fixed as the respective solutions to the above LPs.

4.2. Metrics derived from the stochastic LPs

The solutions to the stochastic LPs provide several useful operational metrics. We first list and

define the metrics we believe are the most useful. We then describe each one separately.

Note that we suppress the item subscript n in the definitions. Additionally, the dummy value

τW can be replaced by cW if cost is being minimized/measured instead of time.


∆≡ V (X, n)

V OPT (χ,n)Balance metric

µ≡∑k

pkdk Weighted average of demand

µ′ ≡∑k

pk min(dk, χ

)Average demand met

γ ≡ µ′

µFraction of demand served

δ≡∑

k:dk≤χ

pk Fraction of disasters served completely

φ≡ V (X)

µ′Average time (cost) per unit delivered

Π Dual variable for constraint (2d) in optimal allocation LP

Π′ ≡Π + (1− δ)τW Adjusted dual variable

πki Dual variable for constraint (1c) for depot i and disaster k

for actual allocation LP

π′i ≡∑k

πki + (1− δ)τW Adjusted dual variable for depot i over all scenarios

To calculate Π requires solving the LP with the optimal inventory allocation, (2a). To calculate

the πi’s requires solving the LP with the actual inventory allocation (1a). To calculate ∆ requires

solving both LPs. The metrics µ′, γ, and δ are the same for both LPs unless θ <∞, in which case

two versions must be calculated for each metric. We also note that µ, µ′, γ, and δ can easily be

calculated without solving any LPs. The stochastic LPs are most useful in calculating Π′, π′i, and

∆.

4.2.1. The balance metric: ∆ The balance metric ∆ is similar to the deterministic version

reported in Acimovic and Graves (2015), which was utilized for an online retailer. The metric is

intended to measure whether a given amount of inventory is generally in the correct place or not.

More specifically, it estimates how far out of balance the actual allocation of inventory is relative

to the optimal.

We note a few properties of this balance metric:

1. It is an approximation of the fractional increase in cost (time) to serve beneficiaries given that

one’s inventory is allocated as it actually is as opposed to being allocated optimally. As such,

if inventory should be in Dubai, but it is actually half way around the world in Panama, the

metric will suffer. However, if the inventory should be in Subang, and it is actually less than

an hour away in Kuala Lumpur, the balance metric value will not change substantially.

2. The optimal value is 1. Anything greater than 1 is considered out-of-balance.


3. It is not strongly affected by abnormally large disasters in the dataset, i.e., it is relatively

robust to outliers. Often, the largest disasters in the set of scenarios (derived from the historical

dataset) far exceed the on-hand supply for any item. The objective functions and solutions of

the LPs above in section 4.1 depend only on those people who are able to be served by the on-

hand inventory. People affected by a disaster who cannot be served by the current inventory

due to a lack of supply have no bearing on the optimal allocation. Thus, if an item can serve

only 1,000 people, then the optimal allocation, the time-to-serve those able to be served, and

other output metrics will be identical whether the largest disaster across the scenarios affects

1,000 people or 10,000,000 people.

4. The metric is affected by which depots are considered. If a new depot is opened in a disaster

hotspot and no inventory is actually moved there, the balance metric will increase (because

V OPT (χ,n) will decrease). In this sense, operational managers can be alerted to the fact that

the inventory is out-of-balance given the new depot.

4.2.2. Fraction served (λ) and fraction of disasters covered (δ) λ represents the frac-

tion of the weighted average of demand met, where the weights are derived from the scenario

probabilities. This value gives a sense as to whether the inventory of items stored in the depots in

total is appropriate. It does not depend on how the items are allocated among the depots (unless

the demand deadline θ <∞). We note that it can be influenced by very large disasters. The inclu-

sion of a disaster scenario affecting tens or hundreds of millions of people will have a significant

impact on µ, the denominator in the fraction that defines λ. Thus, these values in general may

appear low, and must be interpreted with this in mind.

δ, on the other hand, is relatively robust to outliers. If there are only 1000 items in stock of

an item, then whether an unserved disaster affects 1001 or 10,000,000 people is irrelevant in the

calculation of δ. This robustness comes at the cost of not conveying the magnitude of the disasters

that go unserved. δ provides different information from λ, and the two together can help operational

managers understand the adequacy of the total inventory level.

4.2.3. Time per unit delivered The value φ represents the average time (cost) to deliver

one unit to a beneficiary from a depot. This will, of course, not be equal to the actual time to

deliver an item (which would be stochastic itself and would depend on specific factors such as

plane availability and weather), but rather the time from the depot to the capital of the affected

country as calculated using the assumptions and data described below in section 5.3. Even though

this number should not be used to estimate how long it will take for supplies to arrive at a disaster

site, it can provide valuable and objective information when comparing scenarios and strategies

with each other.


4.2.4. Dual variables The increase in total expected time (cost) to serve beneficiaries if an

additional unit were added to depot i can be estimated by the adjusted dual variables (π′i) for

the stochastic LP that utilizes actual inventory allocations (corresponding to constraints (1c)). We

adjust the sum of the original dual variables∑

k πki because they are dependent on the choice of

the dummy value τW (cW ), which is rather arbitrary. That is,∑

k πki reflects the change in the

objective value due to using the dummy node less often and due to the change in total time (cost)

to ship items to beneficiaries from non-dummy node i. The adjusted dual values π′i add back in the

dummy cost multiplied by the fraction of scenarios using the dummy node (1− δ). An additional

unit of an item in the actual system at depot i would alleviate a shipment from the dummy node

in those disasters that are currently utilizing the dummy node. Thus, the adjusted dual variable

excludes the impact an additional unit has on the reliance of the system on the dummy node, and

includes only the impact an additional unit has on the realized time (cost) for those beneficiaries

able to be served.

The value of π′i may be positive or negative. Additionally, the magnitude may be comparable to

the time (cost) to ship one item or it may be considerably less due to the fact that the additional

unit may not be utilized very often. (Note that the dual value reflects the impact on the objective

function, which is an expectation, and it is not conditional on the additional unit being utilized).

For instance, if one were to add a bucket to the current inventory, and if inventory were very low,

then that additional bucket might be utilized in almost every disaster. More beneficiaries would

receive a bucket, and, as such, the objective function V OPT (χ) (with the dummy costs subtracted)

would most likely increase as well. It takes more time in aggregate to serve the beneficiaries, but

only because the system is serving more of them. The magnitude of the dual variable would be

comparable to the average time (cost) to ship a single item. If there were enough buckets in the

system to serve the biggest disaster, then the dummy node is not being used at all, and an additional

bucket might then reduce the overall time to serve beneficiaries from non-dummy nodes; π′i would

then be negative for the depots. The magnitude (absolute value) might be very low due to the fact

that the additional unit is not utilized in most disasters.

To imagine how these dual variables might be used in practice, consider an operational manager

who is deciding whether to add a bucket to the system. She can determine the effect this will have

on fraction of demand served by looking at δ. The additional unit will alleviate a unit of unserved

demand in (1-δ) of the disasters. If this compels the manager to add the unit, then the manager

must decide where. The π′i’s can be used to guide this decision. Assume the manager wants to add

the unit to the depot that results in the smallest increase (or biggest decrease) in total time (cost)

to serve beneficiaries. This can be approximated by the dual variables for each depot. Thus, she

would add it to the depot with the smallest π′i. Additionally, the dual variables may be utilized


to make transhipment decisions. The operations manager might want to move a unit of inventory

from the depot with the largest π′i1 to the depot with the smallest π′i2 . The estimated value of doing

this is π′i1 − π′i2

. If the value of the transshipment exceeds the incurred cost, then the manager

might shift inventory from depot i1 to depot i2.

The adjusted dual variable (Π′) for the stochastic LP that optimally allocates inventory has

similar properties as π′i. However, we have not yet determined the best way to utilize or interpret

this metric.

5. Data

The model described above requires data related to demand, supply, times and costs between

nodes, and item-specific data such as weight and people-served per item. We gather these data

from several distinct sources. We describe our assumptions and the limitations of the data to the

best of our knowledge.

5.1. Disaster data

To generate disaster scenarios, we utilize historical data. As mentioned earlier, our focus is not on

developing forecasting methods and thus we simply project a future that is similar to the past.

Specifically, we assume that each disaster recorded between 1990 until the summer of 2013 has an

equal chance of occurring again, utilizing data from the Emergency Events Database (EM-DAT)

(Centre for Research on the Epidemiology of Disasters 2014). This database keeps track of the

following: the month and year of the disaster, the country, the type of disaster, and the total

affected population (as well as other fields we do not utilize). We make several adjustments to and

assumptions about these data for our analysis of disasters:

1. We use the capital city as the location for any disaster that occurred in a country.

The EM-DAT database does include some data indicating a more precise location,

but this is a text description of where the disaster took place that might be blank

or might be a partial list of provinces, for example. Instead of resorting to judgment

to decipher locations from the few records with such descriptions, we used the geo-

graphic coordinates of country capitals for all historical disasters. From a logistical

perspective, this assumption actually represents a more accurate network for many

countries where the capital is a primary port of entry and/or regulatory hub (e.g.

central medical stores) for imported supplies.

2. We use the “Total Affected” field to measure the number of people affected by a dis-

aster. According to the EM-DAT database’s website description, “Total Affected” is

defined as the sum of “People suffering from physical injuries, trauma or an illness


requiring medical treatment as a direct result of a disaster”, “People needing imme-

diate assistance for shelter”, and “People requiring immediate assistance during a

period of emergency; it can also include displaced or evacuated people,” (Centre for

Research on the Epidemiology of Disasters 2014). Although it can be argued that

this field may be subject to more bias than “Number Killed,” for instance (Peduzzi

et al. 2009), we use it because it is a more accurate representation of the number of

people that need supplies such as blankets, buckets, jerry cans, soap, kitchen kits,

latrine plates, and mosquito nets (the items we evaluate). Additionally, only 20% of

the records have null values for “Total Affected” as opposed to about 27% of null

values in “Number Killed.” We exclude records with null values for “Total Affected”

in our study.

3. We examine only sudden onset disasters and epidemics. Specifically, we include:

earthquakes, epidemics, floods, mass movement dry, mass movement wet, storm,

volcano, and wildfire. We exclude: complex disasters, droughts, extreme tempera-

ture disasters, industrial accidents, insect infestations, miscellaneous accidents, and

transport accidents.

4. We utilize post-1990 data due to completeness and homogeneity. (Peduzzi et al.

(2009) utilize post-1980 data for a similar reason.)

5. Some disasters have a value of 0 for the month. If we are examining disasters and

needs year-by-year, we include these disasters. If we are analyzing the data month-

by-month, we exclude these disasters.

5.2. Depot and warehouse data

Governmental and non-governmental organizations who choose to stockpile disaster relief supplies

may utilize their own warehouses, regional warehouses run by local governments and other organi-

zations, or the United Nations Humanitarian Responds Depots (UNHRDs). We utilize two sources

of data to ascertain the actual amount of supplies at each warehouse. The first is the UNHRD web-

site at http://www.unhrd.org/ (United Nations 2014). This website hosts a real-time stock report

that details which organizations are housing which items where within the UNHRD network. The

second source of data is the “Global Mapping of Emergency Stockpiles” database maintained by

OCHA (UN Office for the Coordination of Humanitarian Affairs 2014b). Organizations volunteer

the following data regarding items in their stockpile: organization name, name of the city for the

stockpile location, item name, and quantity (among some other fields). This database is proprietary

and not open to the public.

If an organization houses an item in a UNHRD, it may show up in both databases. Thus, we

merged them as follows. For each organization, for each item, we assume that the maximum of the


UNHRD number of items and the OCHA number of items is the true number of items owned by

that organization. Also through the data-cleaning process, we merged the names of items that were

similar. For instance, a jerry can may be referred to as “Jerry can, collapsible,w/screw cap,10lt,”

“Jerry can, collapsible, 10 L with Tap cab,” “Jerrycan 20 ltr,” among other descriptions. We

considered for jerry cans and buckets that they were all the same regardless of description or size.

We performed similar data-cleaning on the other items that we examine. Thus, a record consists

of the organization, city, item name, and quantity. Additional data cleaning was required to match

the names of the organizations among the two databases as well. See below in section 5.4.3 for

more details on blankets.

Finally, we took steps to “merge” similar depots. For instance, a depot is listed in Kuala Lumpur

as well as Subang. These locations are about a 30 minute drive from each other. Including both

locations in our model resulted in different inventory allocations in Kuala Lumpur and Subang, even

though the inventory in Malaysia stayed constant. Therefore, to make the model and results easier

to interpret, we picked only one depot for each of the countries in our database, and reallocated

all the inventory from the other depots on that country to the one warehouse. Specifically, we

reallocated inventory from:

• Ottowa, Canada to Toronto, Canada

• Saint-Bueil, France to Roissy-en-France, France

• Kuala Lumpur to Subang

• Molde, Kapp, Kolbotn, and Trollasen, Norway to Oslo, Norway

• Lae, Papua New Guinea to Port Moresby, Papua New Guinea

• Barcelona, Spain to Madrid, Spain

• Gloucestershire, UK to Oxfordshire, UK

5.3. Time and cost data

As just outlined, we have determined the locations of the warehouses and disasters. We now describe

how we calculate the distance, time, and cost from each warehouse to each disaster site. We calculate

these for two modes of transportation: air and truck. We do not consider sea transportation as the

study focuses on immediate deployment of the stockpile. As evident from the LP formulations, we

assume the cost of transporting goods in linear in the number of units being transported. Thus,

for each warehouse-disaster-mode triplet, we calculate the time and the cost-per-metric-ton-km for

the route. This, paired with information on the weights of each item, allows us to calculate the

cost of shipping a single unit on a specific route.

For air, we assume the time and cost are based solely on distance. In order to calculate the

distance between a warehouse and a disaster, we assume that planes travel on a great circle around


the globe starting at the warehouse and landing in the capital city of the country in which the

disaster took place.

For trucks, in order to calculate the distance and time between a warehouse and a disaster, we

utilize Google’s “Distance Matrix” API (application programming interface) (Google 2014). This

provides the duration and distance between two points based on driving. Although we can prefer

to not use ferries with this API, if a ferry route exists, it will return the distance using a ferry. If

there is no way to drive between two points and no ferry exists, no result will be returned, and we

assume air must be used. We believe this is reasonable because if a ferry route exists, organizations

may use this route (or a similar one) to transfer goods via boat.

Having calculated the distance for truck and air, we then calculate the time and cost assuming

there is a fixed and variable component. For cost, we assume that moving a ton of goods incurs

10USD per metric ton fixed cost via truck and 25USD per metric ton fixed cost via air. Additionally,

we assume air incurs 0.50USD per metric ton per km while truck incurs 0.10USD per metric ton

per km.

To calculate the time for air, we assume that airplanes used by these organizations travel at

600 km/hr and that there is a 6 hour fixed time component to secure an airplane. For truck, we

utilize the Google API travel times for driving, if driving is possible. We exclude driving routes

with a driving time greater than 100 hours, assuming that air is the only feasible mode for such

warehouse-disaster arcs. Of the 7175 warehouse-disaster arcs, about 31% are drivable according to

the Google API. Of this subset of drivable routes, 71% take less than 100 hours to traverse.

We approximated the above time and cost parameters based on data spanning numerous com-

mercial and humanitarian projects conducted at the MIT Center for Transportation & Logistics

and on conversations with humanitarian logisticians. As such, the parameters may vary among dif-

ferent organizations working in different contexts in different locations. Our goal is not to exactly

measure the time and cost, but rather to develop values that are relatively precise among the

different routes and modes to enable tactical decision making and that are normative to provide

strategic insight.

5.4. Item specific data

We concentrate on seven specific items that are stored at depots: blankets, buckets, jerry cans,

kitchen sets, latrine plates, mosquito nets, and soap. These non-food items (NFI) are among the

most common items sent to disasters by organizations and consequently have higher inventory

levels at depots. Additionally, these items are different from each other in a way that necessitates

modeling them differently. The need for blankets in a country depends on that country’s weather

as well as the time of year the disaster occurred. We assume mosquito nets are required only in


countries that experience malaria. The other five items may be needed in all disasters, regardless

of location and time of year. For each of these items, we gather weight and cubic volume data as

well as data related to how many beneficiaries each item can serve.

5.4.1. Item weights For each of the items listed above, we accessed the International Fed-

eration of the Red Cross and Red Crescent (IFRC) Societies web interface that catalogues and

describes many of the disaster relief supplies utilized by the IFRC. This website (International

Federation of the Red Cross and Red Crescent Societies 2009) provides information on the weight

and cubic volume of common items. We focus on item weights since most transportation tariffs are

based on weight or ”dimensional weight.”

5.4.2. Items needed per person In section 4.1 we defined a parameter β that dictates for

an item how many units are required per person affected. In reality, this could vary by time of

year, country, temperature, disaster type, and environment. For buckets, jerry cans, kitchen sets,

latrine plates, and soap, we assume that the items required per person are constant and do not

depend on these factors. We assume that need for blankets does vary by country and month within

the year, and that mosquito nets are needed only in countries where malaria is present.

We consulted the Sphere handbook (The Sphere Project 2014), historical appeals for funding that

specify commodity requirements, and humanitarian experts (Bauman 2014) in order to estimate

the items needed per person. For the non-country and non-weather related items we assume that:

1. A bucket serves a family of five (regardless of size of bucket)

2. Two jerry cans serve a family of five (regardless of size of jerry cans)

3. A kitchen set serves a family of five

4. A latrine plate serves 50 people

5. Two mosquito nets (if they are necessary) serve a family of five

6. A bar of soap serves one person

We determined the need for mosquito nets within a country based on whether that country had

risk for malaria. We consulted the CDC website (Centers for Disease Control and Prevention 2014)

to determine which countries had any risk for malaria at all. We assume that if a country has at

least a partial (or localized) risk, then mosquito nets are needed for any disaster that strikes in

that country. Otherwise, no mosquito nets are needed for that country. We do not attempt to track

each country’s disaster month relative to the rainy/malaria season. We assume that the impact of

any disaster is great enough and long-lasting enough that families will need mosquito nets at some

point over the next year following a disaster, which will include the rainy season.


5.4.3. Blankets Blankets are one of the more complicated items because demand depends

on the type of blanket and the climate by region and by month. To calculate how many blankets

are required in a disaster, we interpolate the number of blankets per person required based on

estimating the required number at different temperatures. Implicit in this method is the assumption

that we can serve a fractional number of blankets to beneficiaries on average. That is, if at a low

temperature we serve two blankets per person, and at a high temperature we serve one blanket per

person, then we assume between these temperatures we serve 1.5 blankets on average per person:

half the people require one blanket and half require two.

We calculate the blanket needs per person by first calculating the thermal insulation needs per

person according to temperature. Humanitarian organizations use TOG (thermal resistance of

garments) as a measurement of a material’s ability to insulate. According to guidelines set by the

United Nations High Commisioner for Refugees (2012), thermal insulation requirements per person

are 4 TOG at 10◦C, 6 TOG at 0◦C, 8 TOG at -10 ◦C, and 9 TOG at -20 ◦C. The guidelines also

suggest that someone resting indoors at 20 ◦C requires 1.5 TOG.

Assuming that garments’ and blankets’ abilities to insulate are additive, then a person sleeping

outside in a location with a nightly low of 0 ◦C would need to be provided 6 TOG of insulation

combined; this could come from clothing and blankets, for instance. Assuming that trousers and

a long-sleeved shirt provide about 1 TOG of insulation (American Society of Heating, Refriger-

ating and Air-Conditioning Engineers 2013) implies that 5 TOG remain to be provided through

blankets. A medium weight thermal blanket has 2.5 TOG (United Nations High Commisioner

for Refugees 2012). Thus, two medium weight blankets per person would be required to provide

adequate insulation.

The dominant type of thermal blanket stored in humanitarian organizations’ stockpiles are

medium weight blankets (which we discuss further below). Using the above assumptions and facts,

we regress number of medium blankets required in addition to basic clothing against temperature

Fahrenheit. The relationship (using the five data points of TOG versus temperature Celsius from

United Nations High Commisioner for Refugees (2012)) is essentially perfectly linear (R2 = 0.993),

which is confirmed by visual inspection. The resulting equation that we use is

NumBlanketsPerPerson= (3.34− 0.044(NightlyLowTempInF ))+, (3)

where (a)+ ≡max(a,0). To obtain PeopleServedPerBlanket (the β parameter in our model), we

take the inverse of NumBlanketsPerPerson. Note that PeopleServedPerBlanket is not linear

in temperature. If NumBlanketsPerPerson= 0, we assume that no blankets are needed, and set

the demand to zero.


The above equation is a function of the nightly low temperature. Because we defining demand

at the country level, due to paucity of more specific location data, we assume that the nightly low

temperature is the same across an entire country for the entire month. We captured the average low

temperature by month for each country in our database by querying the website World Weather

Online (2014). This website averages all the cities within a country for which it has data across

all the years for which it has data. Thus, between the temperature data and equation 3, we can

estimate the number of people served per blanket for any country for each month.

To determine how many blankets are available in the stockpile warehouses, we first need to

determine the TOG of the different types of blankets that are stored. We concentrate only on ther-

mal blankets, ignoring cotton blankets (which are sometimes deployed to warm weather disasters).

Table 8 in appendix A lists all the different descriptions of blankets that we found, and our estimate

of the TOG based on UN High Commisioner for Refugees (2013) and International Federation of

the Red Cross and Red Crescent Societies (2009) guidelines. We set the TOG for lightweight blan-

kets to zero because we are excluding them from this analysis. 94.6% of all the thermal blankets

stored in the warehouses in the UNHRD and OCHA databases are medium weight with TOG of

2.5. Instead of determining the blanket need based on heavy and lightweight thermal blankets in

addition to medium weight blankets, we instead assume that 100% of the thermal blankets stored

are medium weight with TOG of 2.5. However, the model could easily incorporate blankets having

different insulating abilities.

6. Results

Having described the LP formulations in section 4.1, we utilize the data described in section 5 to

report on the metrics outlined in section 4.2. We also report on the optimal allocation of inventory,

and how this changes by month and by amount of inventory stored in the system. Unless stated

otherwise explicitly, all results are calculated assuming that there is no delivery deadline, that is,

θ =∞. Additionally, unless otherwise noted, demand refers to number of units required, not the

number of people requiring the units.

6.1. Metrics regarding items’ abilities to meet demand

Table 1 describes each item’s ability to meet demand given the current on-hand inventory level.

These data could easily have been calculated knowing the on-hand inventory level for each item

and the the size of each disaster in each scenario. That is, the results are agnostic to the allocation

of demand across the locations.

We notice from table 1 that the fraction of demand served (γ) tends to be small, even when

almost one million blankets are kept in inventory, and even when these blankets can cover all

the demand in 96% of the disaster scenarios in our dataset. This is driven by several disasters


Table 1 Summary metrics for items in actual depots (Ability to meet demand)

Item Units Demand Demand met Fraction of Fraction of(µ) (µ′) demand disasters

served (γ) served (δ)

Blanket 852,563 561,746 75,150 0.13 0.96Bucket 106,844 185,926 21,206 0.11 0.90

Jerry Can 437,530 371,852 61,300 0.16 0.93Kitchen Set 126,143 185,926 23,105 0.12 0.91

Latrine Plate 4,650 18,593 1,321 0.07 0.83Mosquito Net 395,588 428,591 64,386 0.15 0.91

Soap Bar 111,595 929,631 41,289 0.04 0.76

whose reported “Total Affected Population” is in the hundreds of millions. The paramater γ can

be susceptible to outliers, and as such, care must be taken in its interpretation. Also note that the

96% of disasters fully covered, many of them small, result in low average demand met compared

with inventory levels.

Table 1 (and the other results we present as well) utilize units of item demand, i.e., our model

measures how many blankets were requested and delivered. It is possible, for each disaster scenario,

to convert item demand to people in need and people served using the β parameter. For all items

except for blankets, the ratio of people demand to unit demand is identical regardless of the mix

of disasters served. Blanket demand varies by disaster because a blanket may serve three people in

a warmer climate and half of a person in a colder climate. Table 2 shows the connections between

unit demand and people demand.

Table 2 Item demand, people demand, and people served per unit

Item Demand Demand met Demand Demand Met People served People served(units) (units) (people) (people) per unit (all per unit (only

(µ) (µ′) demand, met for demandor not) met)

Blanket 561,746 75,150 949,801 238,804 1.7 3.2Bucket 185,926 21,206 929,631 106,030 5 5

Jerry Can 371,852 61,300 929,631 153,251 2.5 2.5Kitchen Set 185,926 23,105 929,631 115,527 5 5

Latrine Plate 18,593 1,321 929,631 66,066 50 50Mosquito Net 428,591 64,386 1,071,477 160,965 2.5 2.5

Soap Bar 929,631 41,289 929,631 41,289 1 1

Notice that people served per unit varies over all demand and met demand only for blankets,

which is 1.7 and 3.2 respectively. This reflects the fact that the unmet demand tends to involve

disasters in colder-than-average months and locations.


6.2. Metrics regarding quality of allocation of inventory

Part of the utility from the stochastic LP is derived from its ability to compare disaster response

metrics for the current inventory level and allocation with other stockpile deployment strategies,

including the optimal allocation for each inventory level. In this section, we report metrics derived

from the stochastic LP, including ∆ (balance metric) and φ (average time/cost to ship).

Specifically, table 3 shows the results when time is minimized in the objective value of the

stochastic LP. This table also lists the corresponding cost, average distance traveled, and fraction of

units by air. Table 4 shows the same results, except that cost is minimized in the objective function

of the stochastic LP. Average time-per-unit shipped is in boldface type in the table corresponding

to the LP minimizing time, and average cost-per-unit shipped is boldface in the table corresponding

to the LP minimizing cost.

Table 3 Time optimization for actual inventory allocation: summary metrics (the value being optimized is inbold)

Item Balance Average time Average cost Average Fraction ofMetric (∆) to ship an item to ship an item distance units moved

(time) (hrs) (φ) (USD) (φ) traveled (km) by air

Blanket 1.07 15.6 5.13 5,810 98%Bucket 1.15 16.9 2.66 6,530 99%

Jerry Can 1.17 16.4 0.94 6,250 99%Kitchen Set 1.13 16.3 15.60 6,190 99%

Latrine Plate 1.13 17.5 9.35 6,880 100%Mosquito Net 1.14 16.0 1.34 5,990 100%

Soap Bar 1.16 18.6 0.39 7,580 99%

Table 4 Cost optimization for actual inventory allocation: summary metrics (the value being optimized is inbold)

Item Balance Average time Average cost Average Fraction ofMetric (∆) to ship an item to ship an item distance units moved

(cost) (hrs) (φ) (USD) (φ) traveled (km) by air

Blanket 1.15 22.6 4.89 5,870 81%Bucket 1.35 23.5 2.56 6,540 84%

Jerry Can 1.37 25.6 0.89 6,280 78%Kitchen Set 1.27 26.1 14.63 6,210 77%

Latrine Plate 1.25 25.9 8.88 6,880 83%Mosquito Net 1.36 23.6 1.28 6,020 82%

Soap Bar 1.29 27.5 0.37 7,580 79%

From these tables, one can easily check the quality of a given item deployment. For instance,

relative to the optimal allocation, blankets seem to be the best allocated item for both time and


cost, whereas jerry cans seem to be the worst allocated. Items are significantly more out of balance

with respect to cost than time. Almost all of the items are transported by air when time is being

minimized, whereas about 80% of the items are transported by air when cost is being minimized.

This makes sense because trucks are significantly cheaper than airplanes, and they will be used

more often when cost is an important factor to consider.

Additionally, tables 3 and 4 show a cost-time tradeoff that we can analyze by plotting the efficient

frontier of the competing objectives. To do this, we minimize cost in the stochastic LP while adding

a constraint restricting the total time be less than one of ten values. The ten values of time are

calculated by first optimizing time, second by optimizing cost and measuring the corresponding

time, and third by choosing ten equally spaced time values between (inclusive) this lower and upper

bound.

The resulting efficient frontier and the actual inventory allocation for blankets are presented

in figure 1. One can see that the current inventory allocation is not pareto optimal: significant

improvement may be possible be reallocating items in the network.. The system can maintain the

same average time-to-respond while dramatically reducing costs. We note that the optimal solution

in the figure for a given point corresponds to an inventory allocation as well as a specific air/truck

mix. Thus, it is possible that for a given solution on the frontier, one depot is shipping some items

by air and some by truck to the same disaster scenario. This might actually occur in an actual

situation, as organizations send some units quickly to alleviate immediate needs, while trucking in

the remainder in order to cut costs.

6.3. Allocation strategies

To explore allocation strategies, we focus in depth on one item: blankets. We examine the optimal

allocation of blankets in the network as it varies with: value being optimized (time or cost), total

inventory in the system, and time of year. We also report on the dual variables for the depots

given the current allocation of blankets, which can help guide operations managers as to what

transshipments may be worthwhile.

We first report how fraction of demand served (γ), fraction of disasters served completely (δ), and

average time-to-serve (φ) vary as the inventory level is increased. In figure 2, we chart how as more

blankets are held in the system, more demand is met and more disasters are served completely. It

is not surprising that fraction of disasters served grows more rapidly than fraction of demand met;

as discussed earlier, the magnitudes of very large disasters’ unserved demands will affect fraction of

demand met. We also observe that as more inventory is added to the system, the average time-to-

respond also decreases. It is clear from this chart that additional inventory can serve two purposes:

it can be available to serve more beneficiaries and it can reduce the overall time-to-respond (as


0%

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Efficient frontier of cost versus time

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Figure 1 Blankets: Efficient frontier of cost versus time. Total time is added as a constraint in the LP, and cost

is minimized across ten values for time. The lower bound on time is equal to the LP objective value

when time is minimized. The upper bound on time is equal to the LP objective value when cost is

minimized.

well as the cost). The chart also provides intuition for decision makers in determining the “sweet

spot” inventory level to meet their objectives.

Figure 3 shows how the actual allocation of 852,563 blankets compares to the optimal allocation

of blankets when both time and cost are optimized. The data supporting this as well as data for

buckets and mosquito nets can be found in appendix B in tables 9, 10, and 11 respectively.

This figure suggests that inventory should be moved from Dubai and Ankara to Subang in order to

minimize response time and costs. Optimally, it is better to have items in Ankara when minimizing

time and in Warsaw when minimizing cost. Ankara is nearer to potential disaster locations via air

transportation (based on our assumptions), while Warsaw is better connected to potential disaster

locations via truck.

Figure 3 shows the optimal allocation of blankets given the current inventory level. We extend

this to test several levels of total inventory in figure 4, which shows the optimal allocation of

inventory when time is minimized. (Similar figures for jerry cans and mosquito nets are shown in

appendix C in figures 11 and 12 respectively.)

From figure 4 we see that Dubai and Ankara are good places to put inventory if there is little

inventory in the system. As inventory is added to the system, Subang becomes increasingly impor-

tant because of the larger disasters that occur in Asia. The dotted line on this figure corresponds


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Figure 2 Blankets: Fraction of demand met, fraction of disasters completely served, and average time-to-respond

versus system inventory level (optimal allocation, minimize time)

0 100,000 200,000 300,000 400,000 500,000 600,000 700,000 800,000 900,000

Optimal Cost

Optimal Time

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Subang

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Figure 3 Actual and optimal allocation of blankets across depots


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Dubai

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NairobiJakarta Warsaw Miami Stockholm Panama

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Figure 4 Blankets: Optimal allocation across depots versus total inventory level (minimize time)

to the actual total inventory level of blankets. As such, the allocation of items to warehouses along

this dotted line matches the middle stacked bar in figure 3. We note in figure 4 that one is not nec-

essarily decreasing the amount of inventory in (for instance) Ankara as the total inventory increases

from about one million to fifty million. Rather, the proportion of inventory kept in Ankara reduces,

even if the actual inventory stays constant or increases. This effect is especially pronounced because

the x-axis uses a logarithmic scale.

One interesting thing is that Stockholm is a useful location to store blankets, even when mini-

mizing time. This is due to the fact that because of flight paths on the spherical earth, Stockholm is

nearer to disaster locations like China than other possible depots. In the scenario when one million

blankets are kept in the system, it is optimal to keep about 3% of these blankets in Stockholm.

In the optimal solution, Stockholm serves disasters in China most often, followed by Mexico, and

then Russia.

The optimal inventory allocation may also vary with month. There are two reasons. First, the

regional prevelance of disasters changes by time of year. There is a hurricane season, a rainy season,

a tornado season, and a wildfire season for certain locations. Second, for blankets specifically, as

the temperature changes in different countries throughout the year, a different number of blankets

is required per person in the same location depending on the time of year.

We present the optimal allocation of one million blankets across months when time is minimized

in figure 5. Figure 6 zooms in on depots where less inventory is allocated by showing the same data


in figure 5, but with three primary depots excluded: Dubai, Subang, and Ankara. The optimal

allocation for a given month is calculated by solving the stochastic LP including only demand

for that month plus two months going forward. Thus, the optimal allocation for April includes

disasters that occurred in April, May, and June, while the optimal allocation for December includes

disasters occurring in December, January, and February.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

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Dubai

Subang

Ankara

Nairobi

PanamaJakarta

Miami

Figure 5 Blankets: Optimal allocation across depots by month when one million items are in the system (minimize

time)

From these two figures we see that during summer, the model tends to keep blankets in Asia,

specifically Subang. But also in the summer, Stockholm is an important depot (for reasons men-

tioned previously, namely, that flight paths from Stockholm to Asia are relatively short). As hur-

ricane season picks up in the Caribbean in the fall, more inventory is allocation to Panama and

Miami. As winter approaches, inventory is better positioned in Africa and Turkey.

Reporting on the actual inventory allocation versus the optimal inventory allocation though is

not always helpful to practitioners. If much inventory exists in Panama, and the model states

that “It is optimal to place it in Ankara,” what are the consequences if the operations manager

transships the inventory from Panama not to Ankara, but instead to Stockholm or Barcelona. The

dual variables provide an estimate of the value of an additional unit of inventory in each location.

From this, one can estimate the value of transshiping between locations, or the cost of replenishing

to one depot instead of another due to external considerations such as risk mitigation or political

factors.

Figure 7 shows the adjusted dual variables (π′i) returned by the optimization software.


0%

5%

10%

15%

20%

25%

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Op

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EXC

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: D

ub

ai,

Sub

an

g, A

nka

ra

Nairobi

Panama

Jakarta

GuadeloupeWarsaw

Rio de Janeiro

AccraMiami

Stockholm

Toronto

BrindisiTaren Point

Papua New GuineaBarcelona

Vienna

Figure 6 Blankets: Optimal allocation across depots by month when one million items are in the system (minimize

time) [Zooms in on depots with fewer blankets, excluding Dubai, Subang, and Ankara]

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

NairobiJakarta

DubaiPapua New Guinea

WarsawStockholm

AnkaraVienna

OsloFrankfurt

BrindisiRoissy-en-France

UKRio de Janeiro

BarcelonaToronto

BrisbaneAccra

Taren PointLas Palmas

MiamiGuadeloupe

New ZealandPanama

Subang

Adjusted dual variables ( ): minimize time Adjusted dual variables ( ): minimize cost'i 'i

Figure 7 Blankets: Dual variables for depots for actual allocation while minimizing time and cost

We do not consider or address degeneracy, multiple optimal dual variables, or the validity of the

duals beyond infinitesimal perturbations. The adjusted duals are estimates of the increase in time

and cost in the objective function if an additional unit is place at the specific depot. They are often

positive in the situations we have examined because the total time or total cost to serve might

increase if a unit is added to the system, though this increase offers the benefit of serving more


people. Subang and Jakarta stand out because when adding an item and serving more people the

time and cost actually reduce respectively. This is a product of the fact that the actual inventory

allocations are significantly suboptimal.

The usefulness of the dual variables is derived not necessarily from the specific values of these

variables, but rather from the differences between them. For instance, according to figure 7 the best

transshipment move with respect to time appears to be from Panama to Subang. The estimated

value of moving one unit would be (0.35−−0.03 =) 0.38 total hours. Recall that an additional unit

may not be utilized in every disaster, and the the duals represent the value of an additional unit

over all disaster scenarios, not the value conditional on it being used.

Consider an operations manager who wants to transship from Panama to Ankara, but cannot

for some reason that the model cannot capture. Then, with respect to time, moving the units to

Stockholm instead of Ankara might provide similar value because the associated dual values are

similar. If the operations manager added units, then Subang would be the best place to minimize

time, and Jakarta would be the best place to minimize cost. Thus, in this way, operations managers

can make decisions using the dual variables as a guide. They offer an objective assessment when

considering the many other factors in making decisions such as the political climate, incentives,

risks, and others.

6.4. Delivery deadline cutoff times

The stochastic LP minimizes average time-to-respond. As such, responding to disaster A in 26

hours and disaster B in 7 hours (with an average response time of 16.5 hours) would be preferable

to responding to both in 17 hours. In reality, the relationship between response time and benefit

to the people affected by a disaster may be nonlinear. One can incorporate this in several ways, for

instance, by including a nonlinear objective function, by minimizing the maximum response time,

or by incorporating only disasters that can be served within a delivery deadline cutoff. While many

researchers have studied the most appropriate nonlinear objective function, there is not consensus

yet as to what the objective function should actually be. We felt that minimizing the maximum

response time was not useful here because some remote disasters may be far away from all depots.

Instead, we explore the last option. We set a delivery deadline cutoff value between 6 and 72 hours.

All arcs in the network whose time-to-respond exceeds this cutoff are considered to be infeasible.

Note that because we have set air travel to have a six-hour fixed time to acquire the plane, only

trucks can be used at a six hour cutoff in our model.

Tables 5 and 6 show how the delivery deadline cutoff time affects how much demand can be

served for the seven items in our dataset.

Not surprisingly, given the current allocation of inventory, very little demand can be satisfied

in six hours (only trucks can be used in this time frame due to the assumptions in our model).


Table 5 Number of units (fraction) served within given time window given ACTUAL inventory allocation

Item Delivery deadline cutoff (hrs)6 12 24 48

Blanket1,161 19,153 67,646 75,150

(0.002) (0.034) (0.12) (0.134)

Bucket139 6,366 16,911 21,206

(0.001) (0.034) (0.091) (0.114)

Jerry Can344 20,459 49,815 61,300

(0.001) (0.055) (0.134) (0.165)

Kitchen Set165 7,639 18,829 23,105

(0.001) (0.041) (0.101) (0.124)

Latrine Plate2 236 1,221 1,321

(0) (0.013) (0.066) (0.071)

Mosquito Net337 22,278 53,489 64,386

(0.001) (0.052) (0.125) (0.15)

Soap Bar404 10,866 28,725 41,289(0) (0.012) (0.031) (0.044)

Table 6 Number of units (fraction) served within given time window given OPTIMAL inventory allocation

Item Delivery deadline cutoff (hrs)6 12 24 48

Blanket2,429 23,537 73,496 75,150

(0.004) (0.042) (0.131) (0.134)

Bucket717 8,344 20,632 21,206

(0.004) (0.045) (0.111) (0.114)

Jerry Can1,939 25,191 60,242 61,300

(0.005) (0.068) (0.162) (0.165)

Kitchen Set767 9,235 22,544 23,105

(0.004) (0.05) (0.121) (0.124)

Latrine Plate46 480 1,265 1,321

(0.002) (0.026) (0.068) (0.071)

Mosquito Net2,124 27,001 63,826 64,386

(0.005) (0.063) (0.149) (0.15)

Soap Bar1,520 14,172 39,203 41,289

(0.002) (0.015) (0.042) (0.044)

By allocating the inventory optimally, one can dramatically increase (proportionally that is) the

demand that can be served. By the 48 hour delivery deadline cutoff, every disaster can be served.

Based on our assumptions, the longest flight possible is about 33 hours. Including the fixed time,

the longest flight is about 39 hours. Thus, the delivery deadline cutoff does not affect the number

of people served when it is higher than 39 hours (although the cutoff value may still have an impact

on average time-to-serve and cost-to-serve).

Figures 8, 9, and 10 show how demand served, cost/time per unit, and optimal allocation across

the depots vary with delivery deadline cutoff respectively for blankets. One can see from figure 8

how more demand could be served sooner if inventory were reallocated. The space between the


lines represents the opportunity that exists. For instance, about 5,000 additional blankets could be

delivered within an 18-hour window (on average) if inventory were rearranged in the system.

0

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60,000

70,000

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6 12 18 24 30 36 42

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Delivery deadline cutoff (hrs)

Actual allocation of blankets

Optimal allocation

Figure 8 Blankets: Units of demand served versus delivery deadline cutoff for actual and optimal inventory

allocations

In figure 9 we show how cost-per-unit and time-per-unit vary with the delivery deadline cutoff

given the optimal inventory allocation. We put the fraction of demand served in gray in this figure,

which is the same information as in figure 8. The cost-per-unit does not vary much whether cost

is being minimized or time is being minimized. When the delivery deadline cutoff is small (under

9 hours), the cost is also small because the only short cheap arcs being used; and not very much

of the demand is being met. As the delivery deadline cutoff increases, more demand is met and

the cost increases. At about 30 hours, most of the demand is met. Increasing the time window

beyond thirty hours allows inventory allocations that lead to lower costs as well as the utilization

of cheaper arcs from the same depots (trucks instead of planes where possible).

We see a similar effect in the bottom plot of figure 9 where we report on the time-to-serve.

Similar to cost, the time-to-serve starts off small (because only short arcs are being utilized), then

increases until about 30 hours, then decreases because inventory allocations that lead to quicker

responses on average can be utilized. As the delivery deadline cutoff increases, the average time-

to-serve increases significantly in the model where cost is being minimized. This is due to the fact

that when cost is the objective trucks replace planes where possible with increasing deadline levels.



Frac

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Fraction served

Cost per unit when time is minimized

Cost per unit when cost is minimized

Time per unit when cost is minimized

Time per unit when cost is minimized

Figure 9 Blankets: How time-to-serve, cost-to-serve, and fraction served vary with delivery deadline cutoff

Finally, we show how the optimal allocation of inventory in the network changes with different

delivery deadline cutoffs. Figure 10 shows how the 852,563 blankets should be allocated for each of

the delivery deadline cutoff values. From this figure, we see that the allocation of inventory may be

sensitive to the deadline. For instance, Subang is utilized heavily at 15 hours, but not as much at

12 hours. This is driven largely by the fact that at 12 hours, China is not reachable by any depot

in our database (due to the 6 hour fixed air time and to no depot being within a six-hour flight

of Beijing). At 15 hours, Subang can now reach Beijing, China, within the deadline. There are not

only very large disasters in China, but also many disasters. Putting inventory into Subang at the

15-hour deadline value allows demand in China to be met. (We note that China’s disasters are also

partly driving the jump in demand met between 12 and 15 hours in figure 8).

Similar figures as figure 8 are included in appendix D for jerry cans, mosquito nets, and bars of

soap. These are figures 13, 14, and 15 respectively.

7. Proposed index

We use the results of our empirical study to propose an initial Response Capacity Index (RCI)

that measures the quality of stock deployment in the global disaster response network. This RCI

encompasses information on a variety of aspects that policy analysts and decision makers might

want to consider with a single composite number. We first list attributes that are important to


0

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200,000

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400,000

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6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 60 72

Brisbane

Taren Point

Las Palmas

Barcelona

Brindisi

Frankfurt

UK

Vienna

Oslo

Guadeloupe

Roissy-en-France

Jakarta

Papua New Guinea

Accra

Rio de Janeiro

Panama

Miami

Toronto

Ankara

Nairobi

Stockholm

Warsaw

Dubai

Subang

Subang

DubaiAnkara Dubai

Subang

UK Stockholm

Nairobi

Delivery Deadline Cutoff

Op

tim

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nit

s to

ho

ld in

eac

h d

epo

t

Figure 10 Blankets: Optimal depots to store blankets in versus the delivery deadline. Maximizes satisfied demand

first, then cost

capture in the RCI. For each attribute, we propose a corresponding metric from section 4.2 to

quantify the assessment. The metrics are described in more detail further below: here we merely

allude to the metrics in order to emphasize the attributes themselves. The RCI should incorporate

information on the quality of the following:

1. Location of depots - Are the depots in the right places? That is, if the current inventory were

allocated optimally across the currently available depots, how good is the response? We utilize

“Optimal time to serve” to represent this attribute. This metric (described below in more

detail) will reflect a faster (slower) response time if depots are located in more (less) useful

cities.

2. Allocation of current inventory - Is the inventory in the system spread out among the current

depots efficiently? For this attribute, we utilize the “Balance metric.” This corresponds to two

metrics because we consider both time as well as cost.

3. Amount of inventory - Is there enough inventory in the system? The metric “Fraction of

disasters covered” captures this index attribute.

4. Quick response ability - Can the network response quickly to a significant portion of the

affected population? The metric “Fraction of disasters covered within 12 hours” captures this

attribute. Additionally, this metric captures information on the quality of the depot locations,

the quality of the allocation of inventory, as well as information about the level of inventory.


For each of the seven items we analyze, we convert each of the five metrics into a scale from 0 to

10. The RCI for a specific item is calculated by taking a simple average of these five components.

We take this straightforward approach to normalizing the metrics for two reasons. First, rigorously

determining appropriate weights for the metrics (that are not equal to 1/n) is beyond the scope

of this paper. Second, a simple average is more accessible for users seeking to relate the RCI

components to the single composite number. Other indices such as the National Health Security

Preparedness Index (NHSPI) utilize a similar approach. Specifically, the NHSPI also uses a 0 to

10 scale (as opposed to a 0 to 100 scale) in order to avoid the perception that the number is a

percentage (Association for State and Territorial Health Officials and Centers for Disease Control

and Prevention (2014)); the NHSPI also utilizes simple averages. We envision practitioners and

policy-makers utilizing an item’s RCI value to get an overall sense of the response capacity for the

item and offering the opportunity to drill down to see the values of the individual attributes to

quickly and easily see how the RCI value was calculated.

Each metric is scaled to be between 0 and 10, where 0 is the worst and 10 is the best. To do this

requires that each component have a “best” possible value and a “worst” possible value to anchor

the score. For instance, consider the metric φOPTτ : average time-to-serve if current inventory were

optimally located throughout the system. The value for blankets is φOPTτ = 14.6. We consider the

worst value to be 40 hours, the approximate longest flight possible between a warehouse and a

disaster. We consider the best value to be 6 hours, the fixed time associated with a flight. Thus, we

convert φOPTτ in this way: 10− [(14.6−6)/(40−6)×10] = 7.47. Whether a component is subtracted

from 10 or not depends on whether “higher is better” for that component.

We list here the five metrics that make up the RCI, and our approximate “best” and “worst”

values for each (knowing that with this information, one can calculate the scaled values based on

tables 1, 2, 3, 4, 5, and 6).

1. Average time-to-serve if inventory were optimally allocated (φOPTτ ) - This component mea-

sures the quality of the network structure as a whole as well as the quality of the total inventory

level as it impacts time-to-serve. We choose the worst value to be 40. This is because the

longest distance that might exist between a warehouse and a disaster location is 20,015km

(the radius of the earth times π). Based on our assumptions that airplanes travel at 600km/hr

and that airplanes have a 6 hour fixed time associated with acquisition, when rounded up,

this equates to 40 hours as the longest possible flight time. We consider the shortest possible

time-to-respond to be 6 hours, which is equal to the airplane fixed time of acquisition. Based

on our assumptions, trucks can actually respond quicker than this. However, based on our

results when we optimize time, air is utilized 98%-100% of the time (see table 3).


2. Balance metric: time (∆) - This component measures, based on the network structure and

total inventory level, how well allocated (or balanced) the current inventory is in the system

versus the optimal with respect to time. Because it is a ratio of the objective value based on

the actual allocation to the objective value based on the optimal allocation, the best possible

value is 1. We consider the worst value to be 2. We chose the number 2 because it is a small

integer that it easy to interpret and because of the items we tested, no balance metric for

any item exceeded (or even approached) this value. In practice, it may be possible for an

item’s balance metric to exceed the number 2. In this case, we would scale any balance metrics

greater than or equal to 2 to the value 0 on the 0 to 10 scale.

3. Balance metric: cost (∆) - This is calculated the same as for time.

4. Fraction of disasters completely served (δ) - This component measures how many disasters

can be served given the current inventory level. As mentioned above in section 4.2.2, this

component is relatively robust to very large outlier disasters. The worst possible value is zero,

while the best possible value is 1. Thus, we merely multiply the metric by 10.

5. Fraction of disasters completely served within 12 hours (δ12) - This component reflects the

fraction of disasters able to be completely served given time threshold θ = 12. We then mul-

tiply this value by 10 (since 0 is the lowest fraction of demand that can be served and 1

is the largest). We chose 12 hours because about half the disasters can be served at this

time interval as compared to the number of disasters able to be served completely when no

time constraint exists. Thus, it provides more insight - additional to insight provided by the

non-time-constrained metric δ - into the quality of the network and inventory allocation as

compared to other time thresholds; at 6 hours almost no disasters are covered and at 24 hours

almost all disasters are covered that will eventually be covered.

Table 7 shows the component and final RCI values for the seven items in our dataset.

Table 7 Values of the components (metrics) and composite RCI for each item (in descending order of RCI)

Item Optimal Balance Balance Fraction of Fraction of RCItime to metric metric disasters disasters (simpleserve time cost covered covered in 12 average)

(φOPT ) (∆τ ) (∆c) (δ) hours (δ12)

Blanket 7.47 9.33 8.49 9.60 6.72 8.32Kitchen Set 7.51 8.73 7.25 9.06 5.87 7.68

Mosquito Net 7.66 8.56 6.42 9.13 6.18 7.59Jerry Can 7.63 8.33 6.25 9.29 6.41 7.58

Bucket 7.47 8.46 6.52 8.96 5.16 7.31Latrine Plate 7.23 8.69 7.47 8.28 3.48 7.03

Soap Bar 7.06 8.37 7.07 7.59 4.10 6.84


We make a few observations. The optimal time to serve does not vary much: this is because the

depot locations are the same for all commodities. Only the total inventory amount changes across

items in this analysis. In general, the more inventory the system has the more effectively it can

spread it out and respond quickly. There is more variation for the cost balance metric than for the

time balance metric: this is in part due to the fact that in order to minimize time, airplanes will

be utilized more of the time. In this analysis, obtaining an airplane requires a fixed time of six

hours which dampens differences among the actual flight times across commodities. The RCI as a

whole scores blankets as the best and soap bars as the worst. Depending of the relative importance

of these different items, decision makers might want to dive deeper into the data behind those

commodities with low scores and possibly allocate more resources towards them. For instance,

the soap’s low RCI value is driven largely by the fact that there is not much soap in the system

(compared to other commodities). While there are very many jerry cans in the system, it ranks

in the middle behind kitchen sets and mosquito nets because these jerry cans are not allocated

optimally across the network of depots.

8. Conclusion

By drawing on the techniques of the disaster pre-positioning literature and the intention of the

economic index literature, we develop normalized, useful, objective metrics for the disaster response

community. These metrics are based upon stochastic optimization models often used to decide

inventory placement in disaster response scenarios. We use these metrics to report on the quality

of inventory levels and allocations for seven items, utilizing recent data from the UN on inventory

levels, from em-dat on disasters, and from other sources on commodity attributes and local tem-

peratures. Five of these metrics were combined into our proposed Response Capacity Index (RCI).

From the metrics and the RCI, we communicate qualitatively where budgets might be allocated:

transfers of inventory from one depot to another, additional inventory in the system for specific

items, or even more depots. Dual variables from the stochastic LP provide additional information

to decision-makers as to where inventory transfers might be cost effective.

We also build intuition as to which depots are useful under which conditions; for instance, Dubai

is a good central depot to use when one does not have much inventory; Stockholm is useful from

April to August due to its relatively short flight paths to Asia; Miami and Panama are utilized

more from October to February due to seasonal changes in disaster patterns such as hurricane

season in the Caribbean; over the course of the year, inventory shifts from in general being in Asia

between March and August to being in Anakara and Nairobi the rest of the year. This intuition

can be utilized even as better and more recent data are collected to update the metrics and RCI.

Through these metrics and RCI, we hope to motivate the need for regular, rigorous assessment of

stockpiles. We offer an analytical approach, based on optimal allocation deployed against a detailed


forecast. The inventory data are not completely current, but they are recent and demonstrate the

opportunity to improve the response capacity of current investments by providing these regular,

rigorous assessments. Deploying the current allocation of inventory across depots takes 14% longer

and costs 29% more on average than the respective optimal allocations. Network-wide, bar soap

has low inventory levels relative to demand, as compared to the six other items we analyze. By

intelligently reallocating the inventory currently in the system, and by adding inventory to those

commodities whose value to beneficiaries warrants it, decision-makers can improve the response

capacity of the system.

One difficulty standing in the way of optimizing the system is the fact that currently there are

many decision-makers; each organization (some governmental some not) decides where to put its

inventory in isolation from the decisions of other organizations. This makes measuring response

capacity and improving the system difficult. A first step, which we believe we have achieved with this

research, is to point out the current state-of-the-system as a whole as well as point out opportunities

that can be achieved if organizations do coordinate with each other more. For instance, figure

7 suggests that blankets be moved from Panama to Subang to save money and time. However,

the inventory in Panama is owned not only by several different organizations, but also is owned

by organizations different from those that own blanket inventory in Subang. Before coordination

is embraced by organizations, it is helpful to point out opportunities and the value of working

together.

Not only is it non-trivial for organizations to coordinate with each other; it also takes great effort

to coordinate among NGOs, the private sector, governments, and donors. It is our intention that

these metrics and RCI can be useful for community-wide assessment and coordination. Donors

can see where their dollars will be most effective, and they can hold organizations accountable for

efficiency. Humanitarian logisticians can better deploy stocks because they will know the overall

value of inventory in each specific location. The discussions between logisticians and donors can be

framed by the metrics and RCI for the purpose of working together to assess the capacity and to

adjust investments and deployments dynamically. More organizations can easily be added to the

models. For instance, oftentimes private industry makes commitments to supply commodities to

the disaster community. Sometimes, instead of holding inventory themselves, organizations contract

with private industry to supply a certain amount of inventory in the event of a disaster. These

situations can easily be incorporated into the model, contingent on the data being available.

There are several useful ways to extend this work. First, we aim to engage the humanitarian

community to gather more extensive and more current data. This data might include the following:

better estimates of inventory levels including contracts with private industry; better estimates of

each country’s capacity to respond to beneficiaries for each item in a disaster; and better estimates


of delivery costs and times. Second, currently, we utilize past disaster data to forecast future

needs. There may exist better forecasting methods, such as those used by the insurance sector to

forecast future risk to property. Third, we might extend the research to consider multi-commodity

assessment, incorporating further expert opinion regarding the priority and importance of each

commodity. For instance, if latrine plates and soap bars both have low RCIs, what is the value to

the beneficiaries of allocating an organization’s resources towards one versus the other? Fourth, it

could be enlightening to consider supplier capacity to replenish depots and/or delivery directly to

the affected community. Because of the structure of the stochastic LP, it is natural to incorporate

a supplier level in the network. For instance, if several NGOs have contracts with a single supplier

whose capacity is below the sum of the contracted amounts of the NGOs, the model would highlight

this. Fifth, currently for the RCI we take a simple average of the components of the index. More

sophisticated methods exist. One could develop a more rigorous methodological basis for the index,

incorporating expert opinion to scale and weight the components, consider further attributes, and

leverage further empirical data for testing.

ReferencesAcimovic, J. and Graves, S. (2015). Making Better Fulfillment Decisions on the Fly in an Online Retail

Environment. Manufacturing & Service Operations Management, 17(1):34–51.

American Society of Heating, Refrigerating and Air-Conditioning Engineers (2013). ASHRAE Handbook:Fundamentals. ASHRAE, Atlanta, GA.

Arvis, J. F., Saslavsky, D., Ojala, L., Shepherd, B., Busch, C., and Raj, A. (2014). Connecting to compete- trade logistics in the global economy - the logistics performance index and its indicators. Technicalreport, The World Bank.

Arvis, J. F. and Shepherd, B. (2011). The air connectivity index: Measuring integration in the global airtransport network. Technical Report 5722, The World Bank.

Association for State and Territorial Health Officials and Centers for Disease Control and Prevention (2014).National health security preparedness index.

Bauman, N. (2014). Assumptions for shelter NFI standards. Personal email communication.

Centers for Disease Control and Prevention (2014). CDC - malaria - travelers - malaria information andprophylaxis, by country.

Centre for Research on the Epidemiology of Disasters (2014). EM-DAT: The OFDA/CRED internationaldisaster database. www.emdat.be. Universit catholique de Louvain. Brussels, Belgium.

Davidson, R. and Lambert, K. (2001). Comparing the hurricane disaster risk of u.s. coastal counties. NaturalHazards Review, 2(3):132–142.

de Brito Junior, I., Leiras, A., and Yoshizaki, H. T. Y. (2013). Stochastic optimization applied to the prepositioning of disaster relief supplies decisions in brazil. Technical Report POMS 2013 paper.

Duran, S., Ergun, O., Keskinocak, P., and Swann, J. L. (2013). Humanitarian logistics: advanced purchasingand pre-positioning of relief items. In Handbook of Global Logistics, pages 447–462. Springer.


Duran, S., Gutierrez, M. A., and Keskinocak, P. (2011). Pre-positioning of emergency items for CAREinternational. Interfaces, 41(3):223–237.

Google (2014). The google distance matrix API.

Guha-Sapir, D. and Below, R. (2006). Collecting data on disasters: easier said than done. Asian DisasterManagement News, 12(2):9–10.

Hoffmann, J. (2012). unctad.org | ad hoc expert meeting on assessing port performance.

Hong, X., Lejeune, M. A., and Noyan, N. (2015). Stochastic network design for disaster preparedness. IIETransactions, 47(4):329–357.

International Federation of the Red Cross and Red Crescent Societies (2009). IFRC emergency items cata-logue.

Klibi, W., Ichoua, S., and Martel, A. (2013). Prepositioning emergency supplies to support disaster relief:A stochastic programming approach. Technical Report CIRRELT-2013-19.

Mete, H. O. and Zabinsky, Z. B. (2010). Stochastic optimization of medical supply location and distributionin disaster management. International Journal of Production Economics, 126(1):76–84.

Peduzzi, P., Dao, H., Herold, C., and Mouton, F. (2009). Assessing global exposure and vulnerability towardsnatural hazards: the disaster risk index. Natural Hazards and Earth System Science, 9(4):1149–1159.

Salmern, J. and Apte, A. (2010). Stochastic Optimization for Natural Disaster Asset Prepositioning.Production and Operations Management, 19(5):561–574.

Shapiro, A., Dentcheva, D., and Ruszczyski, A. P. (2014). Lectures on Stochastic Programming: Modelingand Theory, Second Edition. SIAM.

Simpson, D. and Katirai, M. (2006). Indicator issues and proposed framework for a disaster preparednessindex (DPi). Technical Report Working Paper 06-03, University of Louisville/Fritz Institute, Louisville,KY.

The Sphere Project (2014). The sphere handbook.

UN High Commisioner for Refugees (2013). UNHCR - winter conditions adding to hardships for more than600,000 syrian refugees.

UN Office for the Coordination of Humanitarian Affairs (2014a). Emergency stockpiles | OCHA.

UN Office for the Coordination of Humanitarian Affairs (2014b). Global mapping of emergency stockpiles.

United Nations (2014). United nations humanitarian response depot.

United Nations High Commisioner for Refugees (2012). Core relief items catalogue. Technical report, Geneva,Switzerland.

World Weather Online (2014). World weather online.


Appendix

A. Blanket Data

This is a list of all the field in the databases that mention the word blanket. We have converted the text to

TOG per table 8:

Table 8 Insulation (TOG) of blankets of different descriptions


B. Actual and optimal item locations: tables

Table 9 Blankets: actual and optimal allocation across depots

Depot Location Actual Optimal Time Optimal Cost

Accra 24,660 4,183 3Ankara 322,369 88,802 24,615

Barcelona 9,600 0 0Brindisi 41,100 0 0

Brisbane 16,784 0 0Dubai 225,024 269,205 241,213

Frankfurt 2 0 0Guadeloupe 0 881 0

Jakarta 0 9,913 0Las Palmas 12,799 0 0

Miami 3,025 21,440 12,441Nairobi 0 17,791 25,362

New Zealand 500 0 0Panama 75,791 12,335 6,601

Papua New Guinea 600 1,064 644Rio de Janeiro 0 0 2,817

Roissy-en-France 18,960 0 394Stockholm 0 19,571 7,647

Subang 65,156 392,660 451,966Taren Point 4,025 0 0

Toronto 30,000 116 20,109Vienna 2,168 245 0

Warsaw 0 14,357 58,750


Table 10 Buckets: actual and optimal allocation across depots


Accra 19,681 1,374 0Ankara 0 5,980 3,749

Barcelona 0 55 0Brindisi 0 300 842

Brisbane 80 0 0Dubai 59,507 31,578 21,284

Frankfurt 0 0 1,943Guadeloupe 1,000 200 0

Jakarta 0 3,000 0Las Palmas 0 300 0

Miami 0 1,100 800Nairobi 0 1,800 3,000

Panama 16,056 200 0Roissy-en-France 0 0 1,265

Subang 5,520 60,855 73,160Taren Point 0 0 0

Toronto 4,900 0 800UK 100 0 0

Warsaw 0 102 0

Table 11 Mosquito nets: actual and optimal allocation across depots


Accra 34,730 5,566 86Ankara 0 11,108 7,020

Barcelona 5,860 0 0Brindisi 2,000 0 2,055

Brisbane 500 0 0Dubai 206,835 108,456 64,470

Frankfurt 0 0 7,246Guadeloupe 4,530 400 0

Jakarta 0 12,428 0Las Palmas 24,195 434 0

Miami 0 3,600 1,776Nairobi 0 6,541 11,119

Panama 56,148 1,200 400Papua New Guinea 1,600 0 0

Roissy-en-France 0 0 2,004Subang 49,190 245,855 295,988Toronto 10,000 0 3,424


C. Optimal allocations of jerry cans and mosquito nets by inventory levels

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


Perc

ent

of

inve

nto

ry a

lloca

ted

op

tim

ally

New Zealand

Taren Point

Rio de Janeiro

Vienna

Toronto

Papua New Guinea

Panama

Guadeloupe

Stockholm

Barcelona

Las Palmas

Accra

Miami

Warsaw

Jakarta

Brindisi

Nairobi

Subang

Ankara

Dubai


Dubai

Subang

NairobiJakarta Miami Accra Panama

Cu

rren

t in

ven

tory

leve

l: 4

37

,53

0

Jerry cans

Brindisi

Figure 11 Jerry cans: Optimal allocation across depots versus total inventory level (minimize time)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


Perc

ent

of

inve

nto

ry a

lloca

ted

op

tim

ally

Taren Point

Rio de Janeiro

Vienna

Toronto

Papua New Guinea

Panama

Guadeloupe

Stockholm

Barcelona

Las Palmas

Accra

Miami

Warsaw

Jakarta

Brindisi

Nairobi

Subang

Ankara

Dubai


Dubai

Ankara

Subang

NairobiJakarta Miami Accra Panama

Cu

rren

t in

ven

tory

leve

l: 3

95

,58

8

Mosquito nets

Figure 12 Mosquito nets: Optimal allocation across depots versus total inventory level (minimize time)


D. Demand served by hour for jerry cans, mosquito nets, and soap bars

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

6 12 18 24 30 36 42

Un

it d

em

and

ser

ved


Actual allocation of jerry cans

Optimal allocation of jerry cans

Figure 13 Jerry cans: Units of demand served versus delivery deadline cutoff for actual and optimal inventory

allocations

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

6 12 18 24 30 36 42

Un

it d

em

and

ser

ved


Actual allocation of mosquito nets

Optimal allocation of mosquito nets

Figure 14 Mosquito nets: Units of demand served versus delivery deadline cutoff for actual and optimal inventory

allocations


0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

6 12 18 24 30 36 42

Un

it d

em

and

ser

ved


Actual allocation of soap bars

Optimal allocation of soap bars

Figure 15 Soap bars: Units of demand served versus delivery deadline cutoff for actual and optimal inventory

allocations

models, metrics, and an index to assess humanitarian response

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